While hardness of approximating \({\sf D}k{\sf SH}\)  for general arities is relatively less explored, there has been substantial work on lower bounds for \({\sf D}k{\sf S}\) . The strongest known results here are the constant factor inapproximability by Raghavendra and Steurer [1]} assuming SSEH, and the almost polynomial ratio hardness by Manurangsi [2]} assuming ETH. However the techniques from the above works don't apply to our setting since we seek to explicitly quantify the bias approximation curve. Instead, our approach towards establishing Theorem REF  would be to treat \({\sf D}k{\sf SH}\)  as instances of Max-AND subject to a global cardinality constraint. Hence, as is standard, our reduction will use the framework of composing a dictatorship test with an appropriate outer verifier. This is a well studied approach that has been used successfully to show often optimal inapproximability bounds for Max-CSPs (see [3]} for an overview of such reductions).
